Inference for doubly intractable distributions is challenging because the intractable normalizing functions of these models include parameters of interest. Previous auxiliary variable MCMC algorithms are infeasible for multi-dimensional models with large data sets because they depend on expensive auxiliary variable simulation at each iteration. We develop a fast Bayesian indirect algorithm by replacing an expensive auxiliary variable simulation from a probability model with a computationally cheap simulation from a surrogate model. We learn the relationship between the surrogate model parameters and the probability model parameters using Gaussian process approximations. We apply our methods to challenging simulated and real data examples, and illustrate that the algorithm addresses both computational and inferential challenges for doubly intractable distributions. Especially for a large social network model with 10 parameters, we show that our method can reduce computing time from about 2 weeks to 5 hours, compared to the previous method. Our method allows practitioners to carry out Bayesian inference for more complex models with larger data sets than before.
Models with intractable normalizing functions arise frequently in statistics. Common examples of such models include exponential random graph models for social networks and Markov point processes for ecology and disease modeling. Inference for these models is complicated because the normalizing functions of their probability distributions include the parameters of interest. In Bayesian analysis they result in so-called doubly intractable posterior distributions which pose significant computational challenges. Several Monte Carlo methods have emerged in recent years to address Bayesian inferenc…
This paper deals with some computational aspects in the Bayesian analysis of statistical models with intractable normalizing constants. In the presence of intractable normalizing constants in the likelihood function, traditional MCMC methods cannot be applied. We propose an approach to sample from such posterior distributions. The method can be thought as a Bayesian version of the MCMC-MLE approach of Geyer and Thompson (1992). To the best of our knowledge, this is the first general and asymptotically consistent Monte Carlo method for such problems. We illustrate the method with examples from …
Markov chain Monte Carlo methods for exponential family models with intractable normalizing constant, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian…
Intractable posterior distributions of parameters with intractable normalizing constants depending upon the parameters are known as doubly intractable posterior distributions. The terminology itself indicates that obtaining Bayesian inference from such posteriors is doubly difficult compared to traditional intractable posteriors where the normalizing constants are tractable and admit traditional Markov Chain Monte Carlo (MCMC) solutions. As can be anticipated, a plethora of MCMC-based methods have originated in the literature to deal with doubly intractable distributions. Yet, it remains very …
Posterior distributions often feature intractable normalizing constants, called marginal likelihoods or evidence, that are useful for model comparison via Bayes factors. This has motivated a number of methods for estimating ratios of normalizing constants in statistics. In computational physics the logarithm of these ratios correspond to free energy differences. Combining unbiased Markov chain Monte Carlo estimators with path sampling, also called thermodynamic integration, we propose new unbiased estimators of the logarithm of ratios of normalizing constants. As a by-product, we propose unbia…
Longitudinal item response data are common in social science, educational science, and psychology, among other disciplines. Studying the time-varying relationships between items is crucial for educational assessment or designing marketing strategies from survey questions. Although dynamic network models have been widely developed, we cannot apply them directly to item response data because there are multiple systems of nodes with various types of local interactions among items, resulting in multiplex network structures. We propose a new model to study these temporal interactions among items by…
Doubly-intractable distributions appear naturally as posterior distributions in Bayesian inference frameworks whenever the likelihood contains a normalizing function $Z$. Having two such functions $Z$ and $\widetilde Z$ we provide estimates of the total variation and Wasserstein distance of the resulting posterior probability measures. As a consequence this leads to local Lipschitz continuity w.r.t. $Z$. In the more general framework of a random function $\widetilde Z$ we derive bounds on the expected total variation and expected Wasserstein distance. The applicability of the estimates is illu…
Bayesian nonparametric mixture models offer a rich framework for model based clustering. We consider the situation where the kernel of the mixture is available only up to an intractable normalizing constant. In this case, most of the commonly used Markov chain Monte Carlo (MCMC) methods are not suitable. We propose an approximate Bayesian computational (ABC) strategy, whereby we approximate the posterior to avoid the intractability of the kernel. We derive an ABC-MCMC algorithm which combines (i) the use of the predictive distribution induced by the nonparametric prior as proposal and (ii) the…
In this paper, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive non-diagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is …
Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the `cluster centres' is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing…
University of Oxford（牛津大学）；University of Warwick（华威大学）
Bayesian Likelihood-Free Inference methods yield posterior approximations for simulator models with intractable likelihood. Recently, many works trained neural networks to approximate either the intractable likelihood or the posterior directly. Most proposals use normalizing flows, namely neural networks parametrizing invertible maps used to transform samples from an underlying base measure; the probability density of the transformed samples is then accessible and the normalizing flow can be trained via maximum likelihood on simulated parameter-observation pairs. A recent work [Ramesh et al., …
A density estimation method in a Bayesian nonparametric framework is presented when recorded data are not coming directly from the distribution of interest, but from a length biased version. From a Bayesian perspective, efforts to computationally evaluate posterior quantities conditionally on length biased data were hindered by the inability to circumvent the problem of a normalizing constant. In this paper we present a novel Bayesian nonparametric approach to the length bias sampling problem which circumvents the issue of the normalizing constant. Numerical illustrations as well as a real dat…
Indirect inference (II) is a methodology for estimating the parameters of an intractable (generative) model on the basis of an alternative parametric (auxiliary) model that is both analytically and computationally easier to deal with. Such an approach has been well explored in the classical literature but has received substantially less attention in the Bayesian paradigm. The purpose of this paper is to compare and contrast a collection of what we call parametric Bayesian indirect inference (pBII) methods. One class of pBII methods uses approximate Bayesian computation (referred to here as ABC…
Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves the calculation of an intractable normalizing constant. This barrier motivates the consideration of tractable approximations to the likelihood function, such as the pseudolikelihood function, which offers an approach to constructing such an approximation. Naive implementation of what we term a pseudo-posterior resulting from replacing the likelihood function …
Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable, leading to extensive study of approximation methods. Iterative variational methods are a popular and successful family of approaches. However, even state of the art variational methods can return poor results or fail to converge on difficult instances. In this paper, we instead consider computing the partition function via sequential summation over variables. …
Contrastive Bayesian Inference for Unnormalized Models
Graduate School of Economics, Keio University（庆应义塾大学经济学研究科）；Faculty of Economics, Keio University（庆应义塾大学经济学部）；The Institute of Statistical Mathematics（统计数学研究所）；Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息理工学系研究科）；RIKEN Center for Brain Science（理化学脑科学研究中心）
In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) Lévy process. Such processes are intractable for simulation except in certain special edge cases, since the Lévy density associated with the GIG process is expressed as an integral involving certain Bessel Functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct c…
TU Kaiserslautern（凯泽斯劳滕工业大学）；MindGarage；Imperial College London（帝国理工学院）；NeuralSpace；Luleå University of Technology（吕勒奥理工大学）
We introduce a novel uncertainty estimation for classification tasks for Bayesian convolutional neural networks with variational inference. By normalizing the output of a Softplus function in the final layer, we estimate aleatoric and epistemic uncertainty in a coherent manner. The intractable posterior probability distributions over weights are inferred by Bayes by Backprop. Firstly, we demonstrate how this reliable variational inference method can serve as a fundamental construct for various network architectures. On multiple datasets in supervised learning settings (MNIST, CIFAR-10, CIFAR-1…
School of Mathematics and Statistics（数学与统计学院）；University of Melbourne（墨尔本大学）
In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via e.g., the test log predictive density, and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduce…
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-…
CP4SBI: Local Conformal Calibration of Credible Sets in Simulation-Based Inference
Department of Statistics, Federal University of São Carlos（统计系，圣卡洛斯联邦大学）；Institute of Mathematics and Computer Science, University of São Paulo（数学与计算机科学学院，圣保罗大学）；Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK（格勒诺布尔阿尔卑斯大学，法国国家信息与自动化研究所，法国国家科学研究中心，格勒诺布尔INP，LJK）
A recent technology breakthrough in spatial molecular profiling has enabled the comprehensive molecular characterizations of single cells while preserving spatial information. It provides new opportunities to delineate how cells from different origins form tissues with distinctive structures and functions. One immediate question in spatial molecular profiling data analysis is to identify genes whose expressions exhibit spatially correlated patterns, called spatially variable genes. Most current methods to identify spatially variable genes are built upon the geostatistical model with Gaussian p…
The National Health and Nutrition Examination Survey (NHANES) is a major program of the National Center for Health Statistics, designed to assess the health and nutritional status of adults and children in the United States. The analysis of NHANES dental caries data faces several challenges, including (1) the data were collected using a complex, multistage, stratified, unequal-probability sampling design; (2) the sample size of some primary sampling units (PSU), e.g., counties, is very small; (3) the measures of dental caries have complicated structure and correlation, and (4) there is a subst…
With the advance of imaging technology, digital pathology imaging of tumor tissue slides is becoming a routine clinical procedure for cancer diagnosis. This process produces massive imaging data that capture histological details in high resolution. Recent developments in deep-learning methods have enabled us to identify and classify individual cells from digital pathology images at large scale. The randomly distributed cells can be considered from a marked point process, where each point is defined by its position and cell type. Reliable statistical approaches to model such marked spatial poin…
With the growth of interest in network data across fields, the Exponential Random Graph Model (ERGM) has emerged as the leading approach to the statistical analysis of network data. ERGM parameter estimation requires the approximation of an intractable normalizing constant. Simulation methods represent the state-of-the-art approach to approximating the normalizing constant, leading to estimation by Monte Carlo maximum likelihood (MCMLE). MCMLE is accurate when a large sample of networks is used to approximate the normalizing constant. However, MCMLE is computationally expensive, and may be pro…
Deriving Bayesian inference for exponential random graph models (ERGMs) is a challenging "doubly intractable" problem as the normalizing constants of the likelihood and posterior density are both intractable. Markov chain Monte Carlo (MCMC) methods which yield Bayesian inference for ERGMs, such as the exchange algorithm, are asymptotically exact but computationally intensive, as a network has to be drawn from the likelihood at every step using, for instance, a "tie no tie" sampler. In this article, we develop a variety of variational methods for Gaussian approximation of the posterior density …
This paper addresses the problem of estimating the Potts parameter B jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on B requires computing the intractable normalizing constant of the Potts model. In the proposed MCMC method the estimation of B is conducted using a likelihood-free Metropolis-Hastings algorithm. Experimental results obtained for synthetic data show that estimating B jointly with the other unknown parameters leads to estimation results…
Bayesian inference for exponential family random graph models (ERGMs) is a doubly-intractable problem because of the intractability of both the likelihood and posterior normalizing factor. Auxiliary variable based Markov Chain Monte Carlo (MCMC) methods for this problem are asymptotically exact but computationally demanding, and are difficult to extend to modified ERGM families. In this work, we propose a kernel-based approximate Bayesian computation algorithm for fitting ERGMs. By employing an adaptive importance sampling technique, we greatly improve the efficiency of the sampling step. Thou…
